Tight bounds for the sensitivity of CDAWGs with left-end edits

TL;DR

This work investigates how the size of a CDAWG changes when a single character edit is applied at the left end of the input string. It develops tight additive bounds on the increase in the number of edges, $e' - e$, for left-end insertions, deletions, and substitutions, with the strongest results for insertions: $AS_{ ext{LeftIns}}(\mathsf{e},n) \le \mathsf{e}$ and a matching lower bound $AS_{ ext{LeftIns}}(\mathsf{e},n) \ge \mathsf{e}-1$, plus near-tight bounds for the other edits. The paper also extends these insights to leftward online construction, proving a quadratic-time lower bound $\Omega(n^2)$ for updating the CDAWG as the string is prepended leftward, both in the plain online and batched settings. Overall, the results establish robust additive sensitivity bounds for CDAWGs under left-end edits and reveal fundamental limits on leftward maintenance algorithms. These findings have implications for CDAWG-based indexing and compression in the presence of edits and errors, and motivate extending the sensitivity framework to arbitrary-position edits and CDAWG-grammar sizes.

Abstract

Compact directed acyclic word graphs (CDAWGs) [Blumer et al. 1987] are a fundamental data structure on strings with applications in text pattern searching, data compression, and pattern discovery. Intuitively, the CDAWG of a string $T$ is obtained by merging isomorphic subtrees of the suffix tree [Weiner 1973] of the same string $T$, thus CDAWGs are a compact indexing structure. In this paper, we investigate the sensitivity of CDAWGs when a single character edit operation (insertion, deletion, or substitution) is performed at the left-end of the input string $T$, namely, we are interested in the worst-case increase in the size of the CDAWG after a left-end edit operation. We prove that if $e$ is the number of edges of the CDAWG for string $T$, then the number of new edges added to the CDAWG after a left-end edit operation on $T$ does not exceed $e$. Further, we present a matching lower bound on the sensitivity of CDAWGs for left-end insertions, and almost matching lower bounds for left-end deletions and substitutions. We then generalize our lower-bound instance for left-end insertions to leftward online construction of the CDAWG, and show that it requires $Ω(n^2)$ time for some string of length $n$.

Figures (6)

• Figure 1: Illustration for $\mathsf{CDAWG}(T)$ of string $T=(ab)^4 c(ab)^3$. Every substring of $T$ can be spelled out from a distinct path from the source $\varepsilon$. There is a one-to-one correspondence between the maximal substrings in $\mathsf{M}(T) = \{\varepsilon, ab, (ab)^2, (ab)^3, (ab)^4 c(ab)^3\}$ and the nodes of $\mathsf{CDAWG}(T)$. The number of right-extensions of $\mathsf{CDAWG}(T)$ is the number $\mathsf{e}(T)$ of edges, which is 9 in this example.
• Figure 2: Illustration for the CDAWGs of strings $T=(ab)^3 abc(ab)^3$ and $T'= bT=b(ab)^3 abc (ab)^3$ with $m = 3$. The omitted edge labels are all $c(ab)^4$. Observe that $\mathsf{e}(T) = 9$ and $\mathsf{e}(T') = 17$, and hence $\mathsf{e}(T')-\mathsf{e}(T) = 8 = \mathsf{e}(T)-1$ with this left-end insertion.
• Figure 3: Illustration for the CDAWGs of strings $T=(ab)^4 abc(ab)^3$ and $T'= T[2..n]=b(ab)^3 c (ab)^3$ with $m = 3$. The omitted edge labels are all $c(ab)^3$. Observe that $\mathsf{e}(T) = 9$, $\mathsf{e}(T') = 14$, and hence $\mathsf{e}(T')-\mathsf{e}(T) = 5 = \mathsf{e}(T)-4$ with this left-end deletion.
• Figure 4: Illustration for the CDAWGs of strings $T=(ab)^4 c(ab)^3$ and $T'= bT[2..n]=bb(ab)^3 c (ab)^3$ with $m = 3$. The omitted edge labels are all $c(ab)^3$. Observe that $\mathsf{e}(T) = 9$, $\mathsf{e}(T') = 15$, and hence $\mathsf{e}(T')-\mathsf{e}(T) = 6 = \mathsf{e}(T)-3$ with this left-end deletion.
• Figure 5: Illustration for the CDAWGs of strings $T_{k,m}=(ab)^3 cab(ab)^4\$$, $bT_{k,m}=b(ab)^3 cab(ab)^4\$$, and $T_{k+1,m}=(ab)^4 cab (ab)^4\$$ with $k = 1, m = 2$.

Theorems & Definitions (37)

• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• Lemma 5
• proof
• Lemma 6